# Electromagnetic induction

# Electro-dynamic loudspeaker principle

### Exemple : See the practical work on the speaker.

### Fondamental : Mechanic and electric equations of the loudspeaker

The following figure represents a device used as a speaker or microphone.

is a permanent magnet of revolution symmetry around axis.

In the air gap is a radial magnetic field.

In the area where the coil moves, attached to the pavilion , the magnetic field is written :

is a system of mass , likely to move along axis.

It can be set in translation movement if it is subject to an exterior force .

It is also subject to dissipative forces of sum , spring forces of sum applied by a system of springs, as well as Laplace's forces of sum applied on .

The pavilion has a total length of wire equal to and carries an intensity .

It is admitted, by neglecting the helicity of , that each element of the wire can be represented in cylindrical coordinates by :

is powered by a voltage source through the circuit.

, and are the total resistance, inductance and capacitance, relatively to the whole of the circuit, included.

can be expressed as a function of , and .

Indeed, if we sum Laplace's forces on several elements of :

The inertia center theorem applied to the pavilion and projected on axis gives the differential equation (M) verified by .

It shows the mechanical behavior of the system :

The mechanic equation (M) of the system is obtained.

In order to obtain the electric equation, , and the induced electromotive force in must be expressed as functions of .

The electromotive force is induced on an element of :

After integration along in the positive direction of :

(We also can obtain this expression for using an analogy with the Laplace rails).

Kirchhoff's law gives the electric equation (E) of the circuit :

### Fondamental : Energy review

Let's compute :

This equation can be written as such :

With :

represents the mechanic energy of the system and its electromagnetic energy.

The previous equation expresses the energy conservation :

The derivative of is equal to the sum of powers provided by the two energy sources of the system : the exterior force and the voltage source .

To this is subtracted the power dissipated via friction forces (or acoustic energy when the speaker emits a noise) or Joule effect.

### Méthode : Study in forced sinusoidal regime

The system is now under a sinusoidal steady state at fixed frequency.

The equations (M') and (E') linking the complex representations of , , and can be written as such :

And :

The complex notation in has been introduced and :

is the complex impedance of the series dipole.

is the mechanic impedance of the mobile pavilion (ratio between Laplace's force and speed).

When it is used as a speaker, is equal to zero, the energy of the system comes from the source of voltage .

The answer to the system is characterized by relations and , given by (E') and (M') :

Hence :

The quantity is introduced :

From an electro kinetic point of view, it is as though, because of the movement in the magnetic field, another electric impedance is added to .

This impedance, called motion impedance, characterizes the electro-mechanic coupling done by the assemblage.

If the frequency is included in , the vibrations of the pavilion will create an acoustic pressure wave which will generate audible sound.

When it is used as a microphone, then is equal to zero.

The energy of the system comes from the sinusoidal force which expresses the action of the pressure forces from a sound wave on .

The answer of the system is then characterized by the function :

The intensity is proportional to the applied force : the electric signal received is faithful to the force.

It will be recorded, treated and reproduced by another speaker.